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Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values. (English) Zbl 1182.35028
Summary: We consider the positive solution of a Cauchy problem for the following \(p\)-Laplace parabolic equation
\[ u_t = \text{div}(|\nabla u|^{p-2}\nabla u)+u^q,\quad p>2,\;q>1, \] and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence of global and non-global solutions of the Cauchy problem. Furthermore, the life span of solutions is also studied.

MSC:
35B33 Critical exponents in context of PDEs
35K65 Degenerate parabolic equations
35B44 Blow-up in context of PDEs
35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35K15 Initial value problems for second-order parabolic equations
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[1] Alikakos, N.D.; Evans, L.C., Continuity of the gradient for weak solutions of degenerate parabolic equation, J. math. pures appl., 62, 253-268, (1983) · Zbl 0529.35039
[2] Deng, K.; Levine, H.A., The role of critical exponents in blow-up theorems: the sequel, J. math. anal. appl., 243, 85-126, (2000) · Zbl 0942.35025
[3] DiBenedetto, E.; Friedman, A., Hölder estimates for nonlinear degenerate parabolic system, J. reine angew. math., 357, 1-22, (1985) · Zbl 0549.35061
[4] DiBenedetto, E.; Herrero, M.A., On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. amer. math. soc., 314, 187-224, (1989) · Zbl 0691.35047
[5] Friedman, A.; Mcleod, B., Blow-up of solutions of nonlinear degenerate parabolic equations, Arch. ration. mech. anal., 96, 55-80, (1987) · Zbl 0619.35060
[6] Fujita, H., On the blowing up of solutions of the Cauchy problem for \(u_t = \triangle u + u^{1 + \alpha}\), J. fac. sci. univ. Tokyo sec. A, 16, 105-113, (1966)
[7] Galaktionov, V.A.; Kurdyumov, S.P.; Mikhailov, A.P.; Samarskii, A.A., Unbounded solutions of the Cauchy problem for the parabolic equation \(u_t = \nabla(u^\alpha \nabla u) + u^\beta\), Soviet phys. dokl., 25, 458-459, (1980) · Zbl 0515.35045
[8] Galaktionov, V.A., Conditions for nonexistence as a whole and localization of the solutions of cauchy’s problem for a class of nonlinear parabolic equations, Zh. vychisl. mat. mat. fiz., 23, 1341-1354, (1985)
[9] Galaktionov, V.A., Blow-up for quasilinear heat equations with critical fujita’s exponents, Proc. roy. soc. Edinburgh, 124A, 517-525, (1994) · Zbl 0808.35053
[10] Galaktionov, V.A.; Kurdyumov, S.P.; Mikhailov, A.P.; Samarskii, A.A., Blow-up in quasilinear parabolic equations, () · Zbl 1020.35001
[11] Galaktionov, V.A.; Levine, H.A., A general approach to critical Fujita exponents and systems, Nonlinear anal. TMA, 34, 1005-1027, (1998) · Zbl 1139.35317
[12] Gui, C.F.; Wang, X.F., Life span of solutions of the Cauchy problem for a semilinear heat equation, J. differential equations, 115, 166-172, (1995) · Zbl 0813.35034
[13] Guo, J.S., Similarity solutions for a quasilinear parabolic equation, J. austral. math. soc. ser. B, 37, 253-266, (1995) · Zbl 0859.35063
[14] Guo, J.S.; Guo, Y.Y., On a fast diffusion equation with source, Tohoku math. J., 53, 571-579, (2001) · Zbl 0995.35035
[15] Hayakawa, K., On nonexistence of global solutions of some semilinear parabolic equation, Proc. Japan acad., 49, 503-505, (1973) · Zbl 0281.35039
[16] Huang, Q.; Mochizuki, K.; Mukai, K., Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values, Hokkaido math. J., 27, 393-407, (1998) · Zbl 0906.35044
[17] Lee, T.Y.; Ni, W.M., Global existence, large time behavior and life span on solutions of a semilinear parabolic Cauchy problem, Trans. amer. math. soc., 333, 365-378, (1992) · Zbl 0785.35011
[18] Levine, H.A., The role of critical exponents in blowup theorems, SIAM rev., 32, 262-288, (1990) · Zbl 0706.35008
[19] Li, Y.H.; Mu, C.L., Life span and a new critical exponent for a degenerate parabolic equation, J. differential equations, 207, 392-406, (2004) · Zbl 1066.35047
[20] Luckhaus, S.; Dal Passo, R., A degenerate diffusion problem not in divergence form, J. differential equations, 69, 1-14, (1987) · Zbl 0688.35045
[21] Mochizuki, K.; Mukai, K., Existence and nonexistence of global solutions to fast diffusions with source, Methods appl. anal., 2, 92-102, (1995) · Zbl 0832.35083
[22] Mochizuki, K.; Suzuki, R., Critical exponent and critical blow-up for quasilinear parabolic equations, Irsael. J. math., 98, 141-156, (1997) · Zbl 0880.35057
[23] Mukai, K.; Mochizuki, K.; Huang, Q., Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values, Nonlinear anal. TMA, 39, 33-45, (2000) · Zbl 0936.35034
[24] Qi, Y.W.; Levine, H.A., The critical exponent of degenerate parabolic systems, Z. angew. math. phys., 44, 249-265, (1993) · Zbl 0816.35068
[25] Qi, Y.W., Critical exponents of degenerate parabolic equations, Sci. China, 38A, 1153-1162, (1995) · Zbl 0837.35076
[26] Qi, Y.W., The global existence and nonuniqueness of a nonlinear degenerate equations, Nonlinear anal. TMA, 31, 117-136, (1998) · Zbl 0907.35073
[27] Qi, Y.W., The critical exponents of parabolic equations and blow-up in \(R^N\), Proc. roy. soc. Edinburgh, 128A, 123-136, (1998) · Zbl 0892.35088
[28] Weissler, F.B., Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. math., 38, 29-40, (1981) · Zbl 0476.35043
[29] Wiegner, M., A degenerate diffusion equation with a nonlinear source term, Nonlinear anal. TMA, 28, 1977-1995, (1997) · Zbl 0874.35061
[30] Wiegner, M., Blow-up for solutions of some degenerate parabolic equations, Differential integral equations, 7, 1641-1647, (1994) · Zbl 0797.35100
[31] M. Winkler, Some results on degenerate parabolic equations not in divergence form, Ph.D. Thesis, Aachen, 2000
[32] Winkler, M., On the Cauchy problem for a degenerate parabolic equation, Z. anal. anwendungen, 20, 677-690, (2001) · Zbl 0987.35089
[33] Winkler, M., A critical exponent in a degenerate parabolic equation, Math. meth. appl. sci., 25, 911-925, (2002) · Zbl 1007.35043
[34] Zhao, J.N., The asymptotic behavior of solutions of a quasilinear degenerate parabolic equation, J. differential equations, 102, 33-52, (1993) · Zbl 0816.35070
[35] Zhao, J.N., Source type solutions of a quasilinear degenerate parabolic with absorption, Chin. ann. math., 15B, 89-104, (1994) · Zbl 0792.35105
[36] Zhao, J.N., On the Cauchy problem and initial traces for the evolution \(P\)-equation with strongly nonlinear sources, J. differential equations, 121, 329-383, (1995) · Zbl 0836.35081
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